According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W \rbrace$ is an open set in $X$.

My question:

What is the definition of continuity of a multi valued map $\varphi$?

What's the definition of open sets in $\wp(Y)$, in other words, what topology does $\wp(Y)$ have?

$\begingroup$Note that $Y$ is also a space in the statement of the theorem, albeit $Y=X$. What you've written is garbled, as you're talking about $W \subseteq Y$ and also $W \subseteq X$$\endgroup$
– David RobertsMay 20 '13 at 11:50

$\begingroup$Which begs the question (which I think is what the OP might have been getting at): what is the topology on $\mathcal{P}(Y)$ such that $\phi : X\to \mathcal{P}(Y)$ is continuous if and only if its upper- and lower- semicontinuous?$\endgroup$
– Mark GrantMay 20 '13 at 12:00

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$\begingroup$@MarkGrant: If $X$ and $Y$ are Hausdorff, a closed-valued map $\phi\colon X \to \operatorname{Cl}(Y)$ is upper- and lower semicontinuous iff $\phi$ is continuous with respect to the Vietoris topology. A sub-basis for the Vietoris topology is given by $U^- = \lbrace F \in \operatorname{Cl}(Y) \mid F \cap U \neq \emptyset\rbrace$ and $U^+ = \lbrace F \in \operatorname{Cl}(Y) \mid F \subseteq U\rbrace$ where $U$ runs through the open sets of $Y$. [Specialize to $Y$ discrete to get $\mathcal{P}(Y)$]. Gerald points out that upper and lower continuity are more related to order than to a topology.$\endgroup$
– MartinMay 20 '13 at 13:06

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$\begingroup$Mark Grant: It's immediate from the definitions that the relevant topology is the one Martin described. I hadn't known, though, that this is called the Vietoris topology.$\endgroup$
– Steven LandsburgMay 20 '13 at 15:03

$\begingroup$@Martin Thank you very much. Btw, I have found a paper Topologies on Spaces of Subsets by Ernest Michael introduced topology on subsets in its 5th section. $\endgroup$
– Heng GuMay 20 '13 at 16:16

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$\begingroup$Also, this Vietoris topology is the one induced by the Hausdorff metric, in the special case where $X$ is a compact metric space and we take only the closed non-empty subsets (instead of the whole power set). The closed non-empty subsets of a space $X$ in this topology are called the hyperspace of $X$, denoted $H(X)$ or sometimes $2^X$.$\endgroup$
– Henno BrandsmaMay 22 '13 at 4:04

One sensible way of generalizing continuity to set-valued functions (from $X$ to subsets of $Y$) is to require the graph of the function to be closed in the product $X\times Y$. This would be equivalent to the continuity of the function if $Y$ is compact. Thus, the Heaviside function is not continuous because one of the points 0 or 1 on the $y$-axis is not in the graph, but if one redefines it to take both values at 0, the graph becomes closed subset of the plane. See http://en.wikipedia.org/wiki/Closed_graph_theorem for a related (but different) notion.

$\begingroup$The heaviside function can be chosen to be uppersemicontinuous but you can't make its graph closed.$\endgroup$
– Mikhail KatzFeb 24 '14 at 13:46

$\begingroup$To make the graph closed, you make the function set-vauled. All values but one are singletons, but the value at $0$ is the set $[0,1]$.$\endgroup$
– Gerald EdgarFeb 24 '14 at 14:26

$\begingroup$Fine, but this is rather different from the notion of semicontinuity for ordinary functions. By the way, is there a reason to chose the interval [0,1] rather than the two-point set?$\endgroup$
– Mikhail KatzFeb 24 '14 at 15:44

$\begingroup$Yes, it is defferent from the definition for ordinary functions. Some people call it upper hemicontinuity to avoid confusion.$\endgroup$
– alvarezpaivaFeb 24 '14 at 18:08